460 KB88 KB200820252017Conder, Marston D. E.Pisanski, TomažŽitnik, Arjanaletnik:12številka:2str. 383-413COBISSID:18064217ISSN:1855-3966URN:URN:NBN:SI:doc-T8F6W92MenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneaCayleyjev grafkartezični produktkrovni grafločno trazitiven graftip simetrijevozliščno tranzitiven grafVertex-transitive graphs and their arc-types|Let ?$X$? be a finite vertex-transitive graph of valency ?$d$?, and let ?$A$? be the full automorphism group of ?$X$?. Then the arc-type of ?$X$? is defined in terms of the sizes of the orbits of the stabiliser ?$A_v$? of a given vertex ?$v$? on the set of arcs incident with ?$v$?. Such an orbit is said to be self-paired if it is contained in an orbit ?$\Delta$? of ?$A$? on the set of all arcs of v$X$? such that v$\Delta$? is closed under arc-reversal. The arc-type of ?$X$? is then the partition of ?$d$? as the sum ?$n_1 + n_2 + \dots + n_t + (m_1 + m_1) + (m_2 + m_2) + \dots + (m_s + m_s)$?, where ?$n_1, n_2, \dots, n_t$? are the sizes of the self-paired orbits, and ?$m_1,m_1, m_2,m_2, \dots, m_s,m_s$? are the sizes of the non-self-paired orbits, in descending order. In this paper, we find the arc-types of several families of graphs. Also we show that the arc-type of a Cartesian product of two "relatively prime" graphs is the natural sum of their arc-types. Then using these observations, we show that with the exception of ?$1+1$? and ?$(1+1)$?, every partition as defined above is \emph{realisable}, in the sense that there exists at least one vertex-transitive graph with the given partition as its arc-typeNaj bo ?$X$? končen vozliščno-tranzitiven graf valence ?$d$?, in naj bo ?$A$? polna grupa avtomorfizmov grafa ?$X$?. Potem je ločni tip grafa ?$X$? definiran v smislu velikosti orbit stabilizatorja ?$A_v$? danega vozlišča ?$v$? na množici lokov incidentnih z ?$v$?. Za takšno orbito pravimo, da je sebi-prirejena, če je vsebovana v orbiti ?$\Delta$? grupe ?$A$? na množici vseh takšnih lokov grafa ?$X$?, za katere je orbita ?$\Delta$? zaprta za obračanje lokov. Ločni tip grafa ?$X$? je tedaj razčlenitev števila ?$d$? v vsoto ?$n_1 + n_2 + \dots + n_t + (m_1 + m_1) + (m_2 + m_2) + \dots + (m_s + m_s)$?, kjer so ?$n_1, n_2, \dots, n_t$? velikosti sebi-prirejenih orbit, ?$m_1,m_1, m_2,m_2, \dots, m_s,m_s$?, ms pa so velikosti sebi-neprirejenih orbit, v padajočem redu. V tem članku najdemo ločne tipe več družin grafov. Pokažemo tudi, da je ločni tip kartezičnega produkta dveh "tujih si" grafov naravna vsota njunih ločnih tipov. Potem na podlagi teh opažanj pokažemo, da je, z izjemo ?$1+1$? in ?$(1+1)$?, vsaka particija, kot je definirana zgoraj, realizabilna, v tem smislu, da obstaja vsaj en vozliščno-tranzitiven graf z dano razčlenitvijo kot svojim ločnim tipomTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije